In a digital burst mode communication system, a binary alternating preamble normally precedes the data. At a receiver, an oscillator of the same frequency as the transmitted waveform is used, but the phase difference between the received sinusoidal waveform and the oscillator is unknown and needs to be estimated for coherent demodulation. This is known as the carrier acquisition problem.
In a digital demodulator implementation, samples are taken from the in-phase and quadrature components x(t) and y(t), respectively, as shown in FIG. 1. A recovered or estimated carrier is passed through a phase shifter 10 which provides an inphase carrier to mixer 12 and a quadrature carrier to mixer 14. This results in a data stream I=x(t) for the in-phase channel and a data stream Q =y(t) for the quadrature channel, and these are sampled at reference numerals 16 and 18.
The samples are usually taken at a rate equal to or larger than two samples/symbol. Normally, two samples/symbol makes for an efficient implementation, and this is the case considered below. Samples taken on the X channel during the preamble are sequentially numbered as X.sub.1, X.sub.2, X.sub.3, X.sub.4, X.sub.5, . . .
Due to the alternating nature of the binary preamble, and moreover, since the samples are taken at a rate of two samples/symbol, it follows that X.sub.1 =-X.sub.3 =X.sub.5 . . . in the absence of noise. Therefore, in order to decrease the effect of noise, a quantity X.sub.odd is formed by combining the odd-numbered samples on the X channel in the following manner: EQU X.sub.odd =X.sub.1 -X.sub.3 +X.sub.5...
This has the effect of averaging out the value of the odd-numbered samples. The same procedure is repeated to obtain X.sub.even, Y.sub.odd, and Y.sub.even.
Given four values (X.sub.e, X.sub.0, Y.sub.e, Y.sub.0) of 8 bits (1 byte) each, it is desired to find a simple PROM implementation to evaluate ##EQU1## such that the maximum error in evaluating .theta. (because of the finite precision resulting from the finite PROM size) is as small as possible.
This problem arises when implementing a digital demodulator for digital burst mode communication. As noted above, an alternating preamble usually precedes the data. At the receiver, an oscillator of the same frequency as the transmitted waveform is used, but the phase difference, .theta., between the received sinusoidal waveform and the oscillator is unknown, and an estimate of it is desired. As shown in FIG. 1, samples are available from the in-phase and quadrature components x(t) and y(t), respectively, sampled at the rate of two samples/symbol (half a sinusoidal period represents one symbol). By denoting the samples on the X channel as X.sub.1, X.sub.2, X.sub.3, X.sub.4 ..., and the samples on the Y channel as Y.sub.1, Y.sub.2, Y.sub.3, ..., the quantities X.sub.0, X.sub.e, Y.sub.0, and Y.sub.e are formed as follows: EQU X.sub.0 =X.sub.1 -X.sub.3 +X.sub.5 -X.sub.7... EQU X.sub.e =X.sub.2 -X.sub.4 +X.sub.6 -X.sub.8... EQU Y.sub.0 =Y.sub.1 -Y.sub.3 +Y.sub.5 -Y.sub.7... (2) EQU Y.sub.e =Y.sub.2 -Y.sub.4 +Y.sub.6 -Y.sub.8...
where EQU X.sub.1 =cos .theta. sin .alpha.+noise EQU X.sub.2 =-cos .theta. cos .alpha.+noise EQU X.sub.3 =-cos .theta. sin .alpha.+noise (3) EQU X.sub.4 =cos .theta. cos .alpha.+noise
repeats ever four samples and EQU Y.sub.1 =sin .theta. sin .alpha.+noise EQU Y.sub.2 =-sin .theta. cos .alpha.+noise EQU Y.sub.3 =-sin .theta. sin .alpha.+noise (4) EQU Y.sub.4 =sin .theta. cos .alpha.+noise
repeats ever four samples, and where .alpha. is the phase displacement between the sinusoidal signal and the sampling clock.
The purpose of computing X.sub.0, X.sub.e, Y.sub.0, and Y.sub.e as above before estimating .theta. is to decrease the noise variance by averaging out over several symbols before performing squaring operations. Clearly, EQU Y.sub.o.sup.2 +Y.sub.e.sup.2 .apprxeq.sin.sup.2 .theta. EQU and EQU X.sub.o.sup.2 +X.sub.e.sup.2 .apprxeq. cos.sup.2 .theta.
from which it follows that equation (1) is an estimate of .theta. as stated above.
Note that the value of .theta. estimated above will be in the first quadrant (i.e., between 0.degree. and 90.degree.). Therefore, there is a four-fold ambiguity in the value of .theta. that needs to be resolved. This can be taken care of in the detection of a unique word that follows the preamble. It is also possible to reduce the four-fold ambiguity to a two-fold ambiguity (which must then be resolved by the unique word) by examining the, sign of X.sub.0 X.sub.e +Y.sub.0 Y.sub.e.
This method of averaging several symbols before computing an estimate of the carrier phase is well known and widely used. The present invention is directed to finding a simple PROM implementation to obtain an accurate estimate of the carrier phase given four quantities X.sub.odd, X.sub.even, Y.sub.odd, and Y.sub.even.
A method of estimating the carrier phase known in the art is shown in FIG. 2. Basically, the method of the prior art consists of squaring and adding operations performed on X.sub.odd, X.sub.even, Y.sub.odd, and Y.sub.even.
First, (X.sub.e).sup.2 is provided at the output of squaring circuit 20, with the most significant byte being loaded into adder 22 and the least significant byte loaded into adder 24. (-Xo).sup.2 is then provided at the output of squaring circuit 20, and is added to (X.sub.e).sup.2 in adders 22 and 24. The log of (X.sub.e).sup.2 +(X.sub.o).sup.2 is then calculated in log circuit 26. The log of (Y.sub.e).sup.2 +(Y.sub.o).sup.2 is similarly provided by circuits 28-34. The quantity (Y.sub.0.sup.2 +Y.sub.e.sup.2 /(X.sub.0.sup.2 +X.sub.e.sup.2) is then obtained by subtracting the output of log circuit 34 from the output of log circuit 26 in subtracters 36 and 38. Circuit 40 then obtains the square root by dividing by 2, and calculates .theta. by taking the arctan of the result.
The additions are performed with full precision, i.e., 2 bytes, obtained at the adder's outputs because of the desire to obtain an accurate estimate. Next, a division operation is performed. However, in the method of the prior art, the division cannot be accomplished in a PROM since the numerator and the denominator are each 2 bytes long. Therefore, division is accomplished by computing logarithms, subtracting the results, and then taking exponentials, all implemented in PROMs. Finally, an inverse tangent operation is performed to obtain the desired carrier phase estimate. The disadvantage of the method of the prior is that several PROMs and latches (not shown) are required in order to estimate the desired angle.